I've read that Turkish is a very old language, but I can't really find any information on how old Bulgarian is. Which of those two languages is older?
Rather than a direct answer, let me explain why it makes little sense to ask such a question. Current languages didn't appear at a distinct moment in time, but rather it evolved gradually from an older language. For example, Bulgarian (see Bulgarian language#History) descended from Proto-Slavic (also ancestor of other Slavic languages, e.g. Russian), which gradually became Modern Bulgarian. To make classifying things easier, linguistic historians talk about "Old", "Middle" and "Modern Bulgarian", but in truth there is no clear limit between Modern Bulgarian and Middle Bulgarian, or between Old Bulgarian and the previous Slavic language that it descended from. People from each generation speak more or less the same language as their parents, it's only if we "zoom out" we'll notice the difference.
So, what does it have to do with your question? It means that the start of a language is quite arbitrary. There is no objective criteria to say whether 1st century or 10th century is a start of a new period. It depends more on how historians or linguists decide that "Bulgarian" or "Turkish" started at a particular period or century, and does not have any relation to how old a linguistic tradition is.